November 11, 2007

The History of Mathematics as a Teaching Tool

Jones (1969) in Historical Topics for the Mathematics Classroom, of the National Council of Teachers of Mathematics claims that History and up-to-date information about the development of mathematics could a a useful tool for the teacher who wishes to teach "why". Jone's "whys" are of three kinds: chronological, logical and pedagogical. Briefly, chronological whys are about looking at specific facts, (such as the fact that there are 360 degrees in a full turn) or more generally at definitions or axiomatic systems and their development. Logical whys serve into building an understanding of the structure of the axiomatic system through its development and not through its finalised form, which possibly contradicts the historical development as well as "the way perceptions grow in the minds of many of our students" (p.2).  Pedagogical whys "are yhe processes and devices that are not dictated by well-established arbitrary definitions and do not have a logical uniqueness", where history may serve in helping the teacher identify a process during which students use a pedagogical sequence to guide their thoughts and finally achieve an in-depth understanding.

What is very interesting is the fact that Jones often implies that all these tools and pedagogical reccomendations can function in the hands of skillful teachers, who, in their turn have very clear purposes in using these tools and make a very detailed planning for their employment. The author ends up suggesting that "the ingenuity of the teacher" is one of the factors that will determine the approach that may be used. Moreover, knowing history on the behalf of the teacher, may help in clearly distinguishing between old mathematics, newer concepts and end up with an informed perspective on what "modern mathematics" is. 

Taking into consideration the fact that, as Jones mentions, history should be an important component in teachers' education programs, I have come to think that before discussing about employing students' syllabuses based upon history (and culture), the first step shuld be to make sure that the teachers have a good understanding of what they teach, not just as a tool, but mainly as a human procedure that assists in moving the world forward. As the article begins with the words of Bazun, a maths teacher; "algebra is made repellent by the unwillingness or inability of teachers to explain why...", I come to think that he must have a large proportion of rightness in his sayings. Is the modern maths teacher in position to answer to questions such as "why didn't we divide the circle into 100 degrees in the first place?" Probably not. Students perhaps receive this inability and they are therefore lead to the dislike towards something that come to be meaningless in its gist, even for the person who is in charge to teach them that very discipline.

October 27, 2007

Teacher Education through the History of Mathematics

Furingheti's (2007, Educ Stud Math 66:131-143) article on Teacher Education through the History of Mathematics describes a laboratory for prospective mathematics teachers, during which the students familiarised themselves with the history of mathematics. Furinghetti mentions that their view about mathematics teaching almost coincides with what they had experienced as students. The idea was then to reflect on mathematical objects through their history and their construction. In this way, students gain understanding of where do specific notions come from, and what kind of 'adventure' led them where they are today. Lack of understanding of this procedure leads to acquiring mathematical thought and not mathematical thinking (Skemp, 1969, in Furinghetti, 2007). History in this project is more an inspirer rather than a part of the lessons students want to construct per se. In this way, future teachers can understand the roots of the concepts, as it is alleged by many authors.

The students were affected in two modes by this historic element. The 'evolutionary' mode was triggered by the reading of history and allowed the prospective teachers to see how a concept evolved and why they dominated. The 'situated' mode was triggered by reading original historic passages and allowed them to recover the cognitive roots of the concept. What was hoped to achieve through this method was the development of an 'aware way of teaching' on behalf of the new teachers. The side effect that this intervetnion resulted in was that the new teachers were motivated to read and learn history of mathematics. But the main point is that history helps look below the polished formal surface and provides with opportunities for a deeper understanding. 

October 21, 2007

The relationship between Listening and History

Arcavi's and Isoda's article (2007) "Learning to listen: from historical sources to classroon practice" (Educational studies in Mathematics 66:111-129) presents and makes use of the relationship between the skill of listening to the students constructively and reading primary historical sources and uderstanding it. The idea is quite ingenious, since, as they claim, "experiencing the process of understanding the mathematical approach of  a primary historical source can be a sound preparation towards listening to students" (p.116). The basis of this claim has two aspects. The first one is that teachers tend to dismiss answers that are far from expected. By reading mathematics from texts that are several hundreds or event thousands years old, the notation, symbols and presentation are still far from expected but they cannot be dismissed as teachers are well aware of their correctness; they just need more time and attention to search for it. The second aspect that the authors' assumption is based upon is the cryptic character of some of the historical sources; certain tools must be developed and employed by the teacher to make sense of the source. Further questioning, paraphrasis of certain parts and other similar actions play an hermeneutic role in understanding what the text is about. The interepretive character of such seeking is similar to the one necessary for understanding students' answers and therefor potential misconcceptions. I found this using of history very inventive and original. Data used by the authors in the specific article suggest that effective listening improves not only students' underastanding but also teh tecahers' knowledge and ability to make more effective use of it, and history is one way of achieving this important result.

September 15, 2007

Critique of Ethnomathematics

What I am reading now is Vithal's and Skovsmose's Critique of Ethnomathematics. They begin by saying that "we conceive of ethnomathemaitcs and critical mathematics education as two important educational positions in the attempt to develop an 'alternative' mathematics education which expresses social awareness and political responsibility". They consider their critique theoretically as well as in the contaxt of South African education system. They first explore the position of mathematics as a lesson in a modernised society and how "ethnomathematics can be interpreted as a reaction to the cultural imperialism which is built into modernisation theory".

They heavily concentrate upon how "mathematics education can be improved by paying special attention to the ethnomathematics in the culture with which the children (students) are familiar. They appear to think of the following statement as the basic thesis of ethnomathematics: "mathematics education can be improved by considering the cultural background of the students". 

What I belive is that ethnomathematics (and critical mathematics education as well) are about much more than just one's own culture. I feel that thry provide the tools to meet other cultures and - perhaps more importantly -  to meet and understand maths through these cultures. Cultures from the past, current cultures, or 'cultures' of specific groups that use math in a given way (formal, researchers, statisticians, parmacists, farmers and so on).

I would like to see other people's thoughts on that, how do they perceive the utility (or non-utility) of ethnomathematics in education. I hope I will have some comments on that soon. In the meantime, I'll carry on with the related readings and upload any contradictory or agreeing opinions I may come accross with.

August 30, 2007


Writing about web page

Although this webpage is too easy to google, I still provide Wikipedia's definition on Ethnomathematics, the scientific branch that explores the relation between mathematics and culture, how they interact to produce mathematics, as we know it today, as we use it, as it was used, as it is used in different countries, at different times.

Mathematics through History and Culture

Writing about web page

This is just to provide a link to the new webpage created for the lesson Mathematics through History and Culture. This lesson is about to be taught for the first time and comments and/or suggestions are WANTED!! to try to make it better!


The lives of others

DVD front cover
The Lives of Others [DVD] [2007]
5 out of 5 stars
This very recent film is full of sensitivity even though telling a story about very harsh times.

Planet of the Apes

The difference between 1967 Planet of the Apes and 2001 Planet of the Apes is equivalent to the difference between a Thriller and a Scary Movie.

The effects are nothing compared to what directors can do nowdays, but still the message is sound and clear.

Requiem for a dream

DVD front cover
Requiem for a Dream [DVD] [2001]
5 out of 5 stars
All the agony, anxiety and sadness of any sort of addiction. Strong scenes, and marvellous music, the best soundtrack I've ever heard!

My three favorite movies

Writing about web page imdb??

My three favorite movies of all times  - Strongly Reccomended!

Planet of the Apes (1967 version, very important!)

Requiem for a dream (Aronofski, strong language&scenes, but it is worh it)

The lives of others (What we do, is who we are) 

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